Distribution of the thermal neutron

Based on the two-group diffusion theory, the distribution of the thermal neutron flux density for a fast neutron point source in an infinite uniform medium is expressed as [22] ${\phi }_{\text{t}}\left(r\right)=\frac{Q}{4\pi D_{\text{t}}r}\frac{L_{\text{t}}^2}{L_{\text{f}}^2-L_{\text{t}}^2}\left(e^{-r/L_{\text{f}}}-e^{-r/L_{\text{t}}}\right)\mathrm{,}$ (1) where Q is the strength of the fast neutron point source, r is the detector spacing, and D_t, L_f, L_t are the thermal neutron diffusion coefficient, fast neutron slowing-down length, and thermal neutron diffusion length, respectively. In a normal formation, L_f is significantly larger than L_t, and Eq. (1) can be approximated as follows: ${\phi }_{\text{t}}\left(r\right)=\frac{Q}{4\pi D_{\text{t}}r}\frac{L_{\text{t}}^2}{L_{\text{f}}^2-L_{\text{t}}^2}{e}^{-r/L_{\text{f}}}\mathrm{.}$ (2) Assuming that the near and far detector spacings are rs and rl (as shown in Fig. 1, and the detector efficiencies are η_l and η_s, the detector counting ratios of the near and far detectors are expressed as follows: $R_{\text{t}}=\frac{{\eta }_{\text{l}}{r}_{\text{l}}}{{\eta }_{\text{s}}{r}_{\text{s}}}{e}^{\left(r_{\text{l}}-r_{\text{s}}\right)/L_{\text{f}}}.$ (3) The detector counting ratio is related to the length of the fast-neutron slowing-down length.

Fig. 1Full-screen View

Model of point source

In conventional compensated porosity logging, the energy of the neutrons emitted from the Am-Be source is below 5 MeV, and the neutrons are primarily slowed down by elastic scattering with hydrogen in the formation. The counting ratio of the thermal neutrons reflects the porosity of the formation. However, the neutron produced by the D-T reaction has an energy of 14 MeV, and neutrons are slowed down by elastic and inelastic reactions with the elements of the formation. Therefore, the porosity of the formation cannot be directly determined by the counting ratio of thermal neutrons.

The slowing-down model of the fast neutron

In this subsection, a slowing-down model of the D-T neutron is established to investigate the relationship between the slowing-down length L_f and elastic scattering, particularly at low-porosity.

According to the diffusion equation [23], the slowing-down length L_f can be expressed as: $L_{\text{f}}=\sqrt{\frac{\overline{r^2}}{6}},$ (4) where r is the straight-line distance from a D-T neutron slowed down to a thermal neutron. As shown in Fig. 2, the neutron slowing-down process can be calculated using the multigroup calculation method [26].

Fig. 2Full-screen View

Neutron slowing-down process

According to neutronics, $\overline{r^2}=\overline{r_1^2}+\overline{r_2^2}+\dots +\overline{r_n^2}$ (5) where rk is the distance traveled by the neutron between the (k-1)th and kth scattering with the formation, E₀ is the energy of the D-T neutrons of 14 MeV, and E_t is the energy of the thermal neutrons.

When a neuron slows, there are two types of scattering: elastic and inelastic. In elastic scattering, the neutron strikes the nucleus and the state of the nucleus remains in the ground state. The process of inelastic scattering is identical to that of elastic scattering except that the nucleus remains in an excited state. Neutrons are slowed by numerous elastic and inelastic scatterings. Therefore, $\overline{r^2}$ in Eq. (5) is expressed as follows: $\overline{r^2}={\overline{n}}_{\text{in}}\overline{r_{\text{in}}^2}+{\overline{n}}_{\text{el}}\overline{r_{\text{el}}^2}$ (6) where ${\overline{n}}_{\text{in}}$ and ${\overline{n}}_{\text{el}}$ are the average inelastic and elastic scattering numbers, respectively. Further, $\overline{r_{\text{in}}^2}$ and $\overline{r_{\text{el}}^2}$ can be expressed as $\begin{array}{c}\overline{r_{\text{in}}^2}=2{\lambda }_{\text{in}}^2=\frac{2}{{\displaystyle \sum {}_{\text{in}}^2}}\\ \overline{r_{\text{el}}^2}=2{\lambda }_{\text{el}}^2=\frac{2}{{\displaystyle \sum {}_{\text{el}}^2}}\end{array}$ (7) where λ_in and λ_el are the inelastic and elastic mean free paths, respectively, and ∑_in and ∑_el are the macroscopic inelastic and elastic cross-sections, respectively. By substituting Eqs. (6) and (7) into Eq. (4), L_f is rewritten as $L_f=\sqrt{\frac{{\overline{n}}_{\text{in}}}{3{\displaystyle \sum {}_{\text{in}}^2}}+\frac{{\overline{n}}_{\text{el}}}{3{\displaystyle \sum {}_{\text{el}}^2}}}$ (8) A neutron slowing-down model was established to obtain ${\overline{n}}_{\text{in}}$ and ${\overline{n}}_{\text{el}}$. In this model, neutrons were slowed by elastic scattering or both elastic and inelastic scattering. To simplify the calculation, it was assumed that inelastic scattering is effective such that the energy of the neutrons fell below E₁, which is the threshold of inelastic scattering, by single inelastic scattering.

We assume that $a=\frac{{\sum }_{\text{in}}}{{\sum }_{\text{in}}+{\sum }_{\text{el}}}$ and $b=\frac{{\sum }_{\text{el}}}{{\sum }_{\text{in}}+{\sum }_{\text{el}}}$ are the probabilities of inelastic and elastic scattering, respectively, and the neutron energy drops to E₁ after i collisions of elastic scattering. When $i=\frac{\ln \left(\frac{E_0}{E_1}\right)}{\zeta }$, ζ is the lethargy of the formation owing to an elastic collision [23].

Based on this assumption, ${\overline{n}}_{\text{in}}$ and ${\overline{n}}_{\text{el}}$ can be obtained as follows: $\begin{array}{c}{\overline{n}}_{\text{in}}=a\frac{1-b^i}{1-b}\\ =\frac{{\text{Σ}}_{\text{in}}}{{\text{Σ}}_{\text{in}}+{\text{Σ}}_{\text{el}}}\frac{1-b^i}{1-b}\\ {\overline{n}}_{\text{el}}=i+b\frac{1-b^i}{1-b}\\ =\frac{\ln \left(\frac{E_1}{E_{\text{t}}}\right)}{\zeta }+\frac{{\text{Σ}}_{\text{el}}}{{\text{Σ}}_{\text{in}}+{\text{Σ}}_{\text{el}}}\frac{1-b^i}{1-b}\end{array}$ (9) The above equations indicate that ${\overline{n}}_{\text{in}}$ and ${\overline{n}}_{\text{el}}$ are affected by the ∑_el, ∑_in, and ζ, which are determined by the nuclides in the formation. Further, $b\frac{1-b^i}{1-b}$ is the average number of elastic scattering during the neutron slowing-down process from the E₀ to the E₁. As the elastic scattering cross section ∑_el in the formation is considerably larger than the inelastic scattering cross section ∑_in, we have $\frac{1-b^i}{1-b}\approx i=\frac{\ln \left(\frac{E_0}{E_1}\right)}{\zeta }$. By substituting Eq. (9) into Eq. (8), L_f can be rewritten as follows: $L_{\text{f}}=\sqrt{{\left(L_{E_0-E_1}^{\text{in}}\right)}^2+{\left(L_{E_0-E_1}^{\text{el}}\right)}^2+{\left(L_{E_1-E_{\text{t}}}^{\text{el}}\right)}^2}$ (10) where $L_{E_0-E_1}^{\text{in}}=\sqrt{\frac{\ln \left(\frac{E_0}{E_1}\right)}{3\zeta {\text{Σ}}_{\text{in}}\left({\text{Σ}}_{\text{in}}+{\text{Σ}}_{\text{el}}\right)}}$ is the slowing-down length of the inelastic scattering, $L_{E_0-E_1}^{\text{el}}=\sqrt{\frac{\ln \left(\frac{E_0}{E_1}\right)}{3\zeta {\text{Σ}}_{\text{el}}\left({\text{Σ}}_{\text{in}}+{\text{Σ}}_{\text{el}}\right)}}$ is the slowing-down length of the elastic scattering in the neutron slowing-down process from energy E₀ to the E₁ and $L_{E_1-E_{\text{t}}}^{\text{el}}=\sqrt{\frac{\ln \left(\frac{E_1}{E_{\text{t}}}\right)}{3{\displaystyle \sum {}_{\text{el}}^2}\zeta }}$ is the slowing-down length of the elastic scattering through which the neutron is slowed down from the energy E₁ to E_t. Equation (10) shows that the elastic scattering of high-energy neutrons corrects the slowing-down length.

Slowing-down model of the fast neutron

To analyze the effects of elastic and inelastic scattering on the neutron slowing-down length, Table 1 presents the theoretical calculation results for the slowing-down length of D–T neutrons in limestone with different porosities (0% to 100%). Specifically, the threshold of inelastic scattering was set to 3.6 MeV for limestone [24], and the cross sections for inelastic and elastic scattering attained the one correspond to the neutron energies of 14 MeV and10 MeV.

When the porosity of the limestone was 0%, there was no water in the limestone; the macroscopic cross section of inelastic scattering was 0.0279 cm^-1, and the macroscopic cross section of elastic scattering was 0.3248 cm^-1 [24]. Therefore, the free path of inelastic scattering was approximately 35 cm, whereas that of elastic scattering was approximately 3 cm. However, the nuclides in limestone are primarily C, H, and Ca. These nuclides have a weaker ability to slow down neutrons, thus resulting in more elastic collisions (about 165 times). This resulted in ${\left(L_{E_0-E_{\text{t}}}^{\text{el}}\right)}^2=525.56{\text{ cm}}^2$ and ${\left(L_{E_0-E_1}^{\text{in}}\right)}^2=262.36{\text{ cm}}^2$. Thus, the effect of elastic scattering on the slowing-down length exceeded the contribution of inelastic scattering. These factors resulted in a theoretically calculated slowing-down length of the D–T neutrons up to 28 cm. As the amount of water in the limestone increased, the porosity increased. In this situation, the macroscopic cross-section of the inelastic scattering decreased slowly; however, the macroscopic cross-section of the elastic scattering increased rapidly. In particular, hydrogen, the nuclide with the strongest ability to slow down neutrons, also caused the number of elastic scatterings to decrease rapidly. For example, when the porosity was increased to 10%, the number of elastic scatterings decreased to 43, the elastic scattering free path was 2.0 cm, the number of inelastic scatterings decreased to 0.14, and the inelastic scattering free path was 38 cm. At this point, the contribution of the elastic scattering to the slowing-down length was less than that of the inelastic scattering. When the porosity exceeded 50%, the contribution of elastic scattering was significantly lower than that of inelastic scattering and could be ignored.

When the slowing-down power of the formation is weak, neutrons are more likely to reduce their energy through inelastic scattering; therefore, inelastic scattering occurs more frequently. When the slowing-down power of the formation is enhanced, the neutrons can reduce the energy to below E₁ through a few collisions. Therefore, the probability and amount of inelastic scattering decrease. However, the free path of elastic scattering is small, rendering the contribution of inelastic scattering to the slowing-down length more dominant. Table 1 also indicates the contribution of the elastic scattering of the D-T neutron in the high-energy region (14 MeV to E₁). The variation of the slowing-down length of the elastic scattering in the high-energy region ${\left(L_{E_0-E_1}^{\text{el}}\right)}^2$ was consistent with the variation of the whole elastic scattering slowing-down length, that is, the slowing-down length of the elastic scattering decreased rapidly as the porosity increased. However, the slowing-down length of the elastic scattering from E₁ to E_t decreased faster than that in the high-energy region.

When the porosity was 0%, the proportion of ${\left(L_{E_0-E_1}^{\text{el}}\right)}^2$ in the elastic scattering reached approximately 4%. Whereas, at a porosity of 30%, the proportion of ${\left(L_{E_0-E_1}^{\text{el}}\right)}^2$ was approximately 7%. Therefore, the contribution of elastic scattering in obtaining more accurate porosity measurements cannot be ignored. Therefore, when addressing the slowing-down length above the inelastic threshold energy, although the contribution of the slowing-down length primarily originates from inelastic scattering, the contribution of elastic scattering cannot be ignored. Otherwise, it affects the accuracy of the porosity measurement.

In the existing D-T neutron porosity logging method, the elastic scattering of the high-energy part is directly ignored, and porosity measurement is directly associated with elastic scattering below the threshold energy. As shown in the table above, although the elastic scattering of high-energy neutrons has a minimal contribution to the slowing-down length during the D-T neutron slowing-down process, it can still cause an error of approximately 4% if ignored. Therefore, if D-T neutrons are used for porosity logging, it is necessary to establish a suitable porosity logging method suitable for the D-T neutron.

Monte Carlo simulation

To verify the performance of the proposed algorithm, TopMC/SuperMC was used to build the tool model and simulate D-T neutron porosity logging.

With reference to the NeoScope logging tool of Schlumberger, and considering the actual situation of the instrument, including the size of the instrument, detector size, and thickness of the shield layer, the instrument design is shown in Fig. 3. A neutron porosity logging instrument was developed using a D-T neutron source, two ³He neutron detectors, and two shields. The neutron yield of the D-T neutron source was set as isotopic, and the near and far source-to-detector spacings were 35 cm and 70 cm. To reduce the direct measurement interference from the neutron source, neutron shields were placed between the neutron source and the near detector, as well as between the near and far detectors, which were made of tungsten alloy. The borehole, with a diameter of 20 cm, was filled with fresh water and surrounded by limestone rocks of varying porosities with a diameter and height of 200 cm. The instrument was compressed against the borehole wall and the limestone density was 2.703 g/cm³.

Fig. 3Full-screen View

(Color online) Model of the instrument and the formation

Based on the calculation model shown in Fig. 3, the thermal neutron counts were simulated for different porosities. By combining the simulated results with the least-squares method, the coefficients in Eq. (16) are determined as follows: $\varphi =\frac{1}{2}\left(\sqrt{Y^2-4Z}-Y\right)$ (17) In Eq. (17), $Z=0.1431\times f\left(R_{\text{t}}\right)\times \left(g\left(\rho \right)+0.092\right)-0.0046$ and $Y=4.6069-0.9431\times f\left(R_{\text{t}}\right)\times g\left(\rho \right)$.

Verification of the algorithm

According to the algorithm of Eq. (17), the D-T neutron porosity logging was simulated with limestone porosity of 0.1%, 1%, 2%, 3%, 5%, 7%, 10%, 13%, 17%, 21%, 26%, 31%, 37%, 43%, 50%, 57%, 65%, 73%, 82%, 91%, and 99.9%. Figure 4 shows the performance of Eq. (17) for different porosity values, which verifies the accuracy of the equations. The calculated values almost always fell on the line 45°, and it is evident that the porosity values calculated using Eq. (17) were highly consistent with the true porosity values. In addition, Figure 4 shows the relative errors of the calculated porosity values, which indicate that the relative error between the calculated and real porosities was less than 2% when the porosity values exceeded 10%. Further, the absolute error of porosity was less than 0.1% when the porosity values were less than 10%.

Fig. 4Full-screen View

(Color online) Results of porosity and error determined by Eq. (17)

If ${\left(L_{E_0-E_1}^{\text{el}}\right)}^2$ in Eq. (11) is neglected, the slowing-down length can be described as $L_{\text{f}}=\sqrt{\frac{\ln \left(\frac{E_0}{E_1}\right)}{3\zeta {\text{Σ}}_{\text{in}}^2}+\frac{\ln \left(\frac{E_1}{E_{\text{t}}}\right)}{3\zeta {\text{Σ}}_{\text{el}}^2}}$. Thus, Equation (17) is simplified and the algorithm of the model can be obtained as follows: $\varphi =1.768\times \frac{\ln {\left(0.811\times R_{\text{t}}\right)}^2}{\rho }-0.000176$ (18) Figure 5 presents a comparison of the results calculated using Eqs. (17) and (18). As evident, the porosity values calculated using Eqs. (17) and (18) were consistent with the actual porosity values shown in Fig. 5(a). Figure 5(b) shows a comparation of the relative errors obtained using Eqs. (17) and (18) when the porosity values exceeded 10%. Compared with Equation (17), the relative error of Eq. (18) was larger, particularly at low porosities. Table 2 indicates that the absolute error reached 3.6%. for Eq. (18), because the actual porosity was less than 7%. This demonstrates that the influence of elastic scattering from E₀ to E₁ on the porosity measurement cannot be ignored.

Fig. 5Full-screen View

(Color online) Comparison of the results calculated by Eqs. (17) and (18). (a) Calculated porosity and (b) relative error

Influence of density accuracy on porosity measurement

The relationship between density and porosity can be observed from Eqs. (12) and (16). As evident, as the density decreases, the porosity tends to increase. The introduction of density correction has resulted in problems in the accuracy of porosity measurements caused by density measurements. In conventional density measurements, the measurement error, Δρ, should generally be less than 0.015 g/cm³. To evaluate the effect of density precision on the neutron porosity measurement, the calculated results were obtained as shown in Fig. 6 and Table 3 for density errors of +0.015 g/cm³ and -0.015 g/cm³.

Fig. 6Full-screen View

(Color online) Effects of density on porosity for density error of (a) +0.015 g/cm³ and (b) -0.015 g/cm³

As shown in Fig. 6 and Table 3, the absolute error between the calculated and actual porosities was less than 0.3% when the actual porosity was below 7%. Further, the relative error was less than 4% in areas where the actual porosity was above 7%. In particular, when the porosity exceeded 2%, the relative error was less than 30%. As the porosity increased, the relative error gradually decreased. Therefore, the measurement error of the density exerts a minimal effect on the porosity in conventional density measurements.

This method considers the nonlinear relationship between the detector count and porosity and density during the transport of D-T neutrons in the formation, indicating that the proposed algorithm can effectively improve the measurement accuracy of the porosity compared with the existing density correction algorithm. Further, it greatly eliminate the influence of density measurement error in the actual measurement process.